Geometry

high importance~4 Q in Tier 137 formulas⚡ 19 shortcuts6 subtopics

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Lines and angles

⏱ 4 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

Angles on one straight line add to $180^\circ$ . Angles around one point add to $360^\circ$ . When two lines cross, the angles facing each other are equal. When a line cuts two parallel lines, matching angles are equal and one inside pair adds to $180^\circ$ . Almost every question becomes one small equation in $x$ .

The two angle pairs

An angle measures the turn between two lines.

Two angles are complementary if they add to $90^\circ$ .
Two angles are supplementary if they add to $180^\circ$ .

For any angle $x$ : its complement is $90^\circ-x$ and its supplement is $180^\circ-x$ .

Rule: Supplement $-$ complement $=90^\circ$ , always. For $x=50^\circ$ : $130^\circ-40^\circ=90^\circ$ .

If a supplement exceeds its angle by $d$ , the angle is $\dfrac{180^\circ-d}{2}$ . With $d=30^\circ$ : angle $=75^\circ$ , supplement $=105^\circ$ , gap $=30^\circ$ . It works.

One line, one point, two crossing lines

Angles on one straight line add to $180^\circ$ . Angles filling one full turn around a point add to $360^\circ$ .

Two straight lines cross and make four angles:

Vertically opposite angles (facing each other) are equal.
Any two neighbouring angles add to $180^\circ$ .

So one angle of $75^\circ$ fixes all four: $75^\circ, 105^\circ, 75^\circ, 105^\circ$ .

Tip: Given three angles around a point, the fourth is $360^\circ$ minus their sum. That is the whole sum.

Parallel lines cut by a transversal

A transversal is a line that crosses two parallel lines. It makes eight angles, but only two different sizes.

Pair	Shape	Rule
Corresponding	F	equal
Alternate	Z	equal
Co-interior (allied)	C, same side inside	add to $180^\circ$

Watch: Co-interior angles add to $180^\circ$ only because the lines are parallel. No parallel lines, no rule.

Turn the picture into one equation

Write the relation the picture shows. Then solve for $x$ .

Two co-interior angles are $(2x+30)^\circ$ and $(x+15)^\circ$ :

(2x+30)+(x+15)=180 \Rightarrow 3x=135 \Rightarrow x=45

So the angles are $120^\circ$ and $60^\circ$ . Check by adding: $120^\circ+60^\circ=180^\circ$ .

A ratio works the same way. Co-interior angles in the ratio $2:3$ are $2k$ and $3k$ , so $5k=180^\circ$ . The angles are $72^\circ$ and $108^\circ$ .

Watch: After finding $x$ , re-read the question. It often asks the larger angle, the smaller one, or a partner angle, not $x$ itself.

Bisectors

A bisector cuts an angle into two equal halves. Halving keeps every relation, only smaller.

Supplementary angles $110^\circ$ and $70^\circ$ have halves $55^\circ$ and $35^\circ$ . The gap halves too: from $40^\circ$ to $20^\circ$ .

The bisectors of a co-interior pair meet at $90^\circ$ . The halves add to half of $180^\circ$ , so the remaining angle in their triangle is exactly $90^\circ$ .

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Co-interior angles with x in them

How to spot it:

Parallel lines with a transversal; the two inside angles on one side are given as $(ax+b)^\circ$ and $(cx+d)^\circ$ .

(ax+b)+(cx+d)=180^\circ

Method

Add the two expressions and set the total to $180^\circ$ .
Solve the linear equation for $x$ .
Put $x$ back into the angle asked.
Check the two angles add to $180^\circ$ .

Why it works:

Co-interior angles between parallel lines are supplementary, so the expressions must total $180^\circ$ .

Try this

Two co-interior angles between parallel lines are $(2x+30)^\circ$ and $(x+15)^\circ$ . Find the larger angle.

Show solution

Sum: $(2x+30)+(x+15)=180$ .
$3x+45=180$ , so $x=45$ .
Angles: $2(45)+30=120^\circ$ and $45+15=60^\circ$ .

Answer

120°

Type 2very common2 practice Q

Corresponding or alternate angles are equal

How to spot it:

Two angles in an F-shape (corresponding) or Z-shape (alternate) position, both with $x$ in them, or one given and its partner asked.

ax+b=cx+d

Method

Spot the pair shape: F means corresponding, Z means alternate. Both are equal.
Equate the two expressions.
Solve for $x$ and substitute back.
Any neighbouring angle is $180^\circ$ minus this one.

Why it works:

A transversal across parallels makes only two angle sizes, and an equal pair uses the same size twice.

Try this

Two corresponding angles between parallel lines are $(3x-10)^\circ$ and $(2x+15)^\circ$ . Find the angle.

Show solution

Equal pair: $3x-10=2x+15$ .
$x=25$ .
Angle: $3(25)-10=65^\circ$ .

Answer

65°

Type 3very common2 practice Q

Complement and supplement of an angle

How to spot it:

'The supplement of an angle is n times its complement', or a difference between supplement and complement is mentioned.

180^\circ-x=n(90^\circ-x)

Method

Write complement $=90^\circ-x$ and supplement $=180^\circ-x$ .
Turn the sentence into one equation.
Solve for $x$ .
Check with supplement $-$ complement $=90^\circ$ .

Why it works:

Both come from the same $x$ , so any 'n times' sentence is one linear equation.

Try this

The supplement of an angle is three times its complement. Find the angle.

Show solution

$180-x=3(90-x)$ .
$180-x=270-3x$ , so $2x=90$ .
$x=45^\circ$ . Check: supplement $135^\circ$ , complement $45^\circ$ .

Answer

45°

Type 4common2 practice Q

Angles on a line, around a point, crossing lines

How to spot it:

Several angles at a point or on a line are given and one is missing, or two crossing lines give one angle and ask the rest.

\text{line}=180^\circ,\quad \text{point}=360^\circ

Method

Pick the total: $180^\circ$ on a line, $360^\circ$ around a point.
Add the known angles.
Subtract from the total to get the missing angle.
For crossing lines, mark the equal vertically opposite pairs.

Why it works:

The totals are fixed, so the missing angle is always a remainder.

Try this

Two straight lines intersect. One of the four angles formed is $75^\circ$ . Find the obtuse angle among the other three.

Show solution

Neighbours are supplementary: $180-75=105^\circ$ .
The vertically opposite angle is also $105^\circ$ .
Both remaining angles are obtuse: $105^\circ$ .

Answer

105°

Type 5occasional2 practice Q

Angle bisectors in line figures

How to spot it:

An angle is bisected and the halves, or the angle between two bisectors, are asked.

\text{half}=\dfrac{\theta}{2}

Method

Halve every angle the bisector touches.
Halve every difference too: bisecting keeps relations but halves gaps.
Remember: bisectors of a co-interior pair meet at $90^\circ$ .

Why it works:

Cutting two angles in half cuts every gap between them in half as well.

Try this

Two supplementary angles differ by $40^\circ$ . After both are bisected, what is the difference between the halves?

Show solution

The angles are $110^\circ$ and $70^\circ$ .
Halves: $55^\circ$ and $35^\circ$ .
Difference of halves: $55-35=20^\circ$ .

Answer

20°

Formula sheet

Angles on a line / around a point

180^\circ,\ 360^\circ

A straight line totals 180 degrees; one full turn totals 360.

Co-interior angles

a+b=180^\circ

The two inside angles on one side of a transversal, between parallel lines.

Complement / supplement

\text{supplement}-\text{complement}=90^\circ

Complement = 90 − x, supplement = 180 − x.

Equal pairs at parallels

\text{corresponding}=\text{alternate}=\text{vertically opposite}

Each of these pairs is equal.

Shortcuts that save time

⚡ Say the shape, then the rule

Trace the two angles with your finger. An F or Z shape means they are equal. A C shape means they add to $180^\circ$ .

Example

Two co-interior angles between parallel lines are $(3x+20)^\circ$ and $(2x+10)^\circ$ . Find the larger angle.

Show solution

C shape, so the sum is $180^\circ$ : $5x+30=180$ .
$x=30$ .
Angles: $3(30)+20=110^\circ$ and $2(30)+10=70^\circ$ .

Answer

110°

⚡ Supplement minus complement is always 90

Whatever the angle, its supplement and its complement differ by exactly $90^\circ$ . Use it as a free check, or to build the equation.

Example

The supplement of an angle is four times its complement. Find the angle.

Show solution

Let the angle be $x$ . Then $180-x=4(90-x)$ .
$180-x=360-4x$ , so $3x=180$ .
$x=60^\circ$ . Check: supplement $120^\circ$ , complement $30^\circ$ , ratio $4$ .

Answer

60°

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding co-interior angles to $180^\circ$ when the lines are not parallel.

Check the question says parallel first. Only then use the $180^\circ$ rule.

Mistake 02

Mixing up the Z pair and the C pair in the transversal figure.

Z (alternate) pairs are equal; C (co-interior) pairs add to $180^\circ$ . Trace the shape.

Mistake 03

Solving for $x$ and marking $x$ as the answer.

Re-read the last line, then put $x$ into the angle actually asked.

Mistake 04

Writing the complement as $180^\circ-x$ .

Complement is $90^\circ-x$ . Supplement is $180^\circ-x$ .

Mistake 05

Halving one angle but not the other in bisector questions.

A bisector halves both angles, so every difference also halves.

Quick revision

Read this the night before the exam.

On a line: $180^\circ$ . Around a point: $360^\circ$ . Vertically opposite angles are equal.
Parallel lines: F-shape and Z-shape pairs equal, C-shape pairs add to $180^\circ$ .
Complement $=90^\circ-x$ , supplement $=180^\circ-x$ ; they always differ by $90^\circ$ .
Supplement exceeds the angle by $d$ : angle $=\dfrac{180^\circ-d}{2}$ .
Solve the equation for $x$ , then answer the angle asked.
Bisectors halve angles, and halve the gaps between them.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.

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